There exists a direction-of-arrival estimation apparatus that measures two types of angle information such as an azimuth and an elevation at the same time (see, for example, A. Swindlehurst and T. Kailath, “Azimuth/Elevation Direction Finding Using Regular Array Geometories” IEEE Trans. on AES, Vol. 29, No. 1, pp. 145-156, January 1993). Such a direction-of-arrival estimation apparatus, for example, obtains two-dimensional spatial phase information of an incoming signal using a grid array sensor. The direction-of-arrival estimation apparatus performs known calculations on the two-dimensional spatial phase information of the incoming signal and thereby estimates two types of angle information, i.e., an azimuth and an elevation.
FIG. 1 illustrates an exemplary arrangement of sensors of a direction-of-arrival estimation apparatus (hereafter, “sensor” is synonymously used with “antenna”).
In the exemplary arrangement of antennas of the direction-of-arrival estimation apparatus illustrated in FIG. 1, it is assumed that a target exists in the positive direction of the Y-axis. Element antennas are arranged at points on the X-Z plane to form an array antenna (hereafter, those points are represented by lattice points for brevity).
With the origin as the center (although the origin coordinate values are set at (x,y)=(1,1)), there are N coordinate points in the positive direction of the X-axis (N is an integer greater than or equal to 1) and M coordinate points in the positive direction of the Z-axis (M is an integer greater than or equal to 1). That is, there are N×M lattice points in total. In FIG. 1, each element antenna is represented by Anm. Element antennas Anm are not necessarily arranged at regular intervals. When intervals dx and dz are the greatest common divisor among all element-to-element spacing along X and Z axes, it can be assumed that antennas arranged at regular intervals and antennas arranged at random intervals are both arranged on lattice points with the intervals dx and dz, and their positions can be handled in a similar manner.
In FIG. 1, it is also assumed that an angle measured clockwise from the positive direction of the Y-axis is the positive direction of azimuth and an angle measured from the X-Y plane in the positive direction of the Z-axis is the positive direction of elevation. That is, assuming that a source of a radio wave is present at point P in FIG. 1, the azimuth and the elevation of the emitter are indicated by θ and φ.
The direction of arrival of a radio wave to be measured is represented by an angle at which an echo signal (reflected wave), which is generated when a probe signal emitted from an apparatus is reflected by a target, returns to a receive antenna. When the point P is a target position, i.e., the position of the source of an echo signal, the spatial phase of an incoming signal received by an element antenna Anm of the array antenna with reference to A11 is represented by formula (1) below.
                                          2            ⁢            π                    λ                ⁡                  [                                                    (                                  n                  -                  1                                )                            ⁢                              d                x                            ⁢              sin              ⁢                                                          ⁢              θ              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              ϕ                        +                                          (                                  m                  -                  1                                )                            ⁢                              d                z                            ⁢              sin              ⁢                                                          ⁢              ϕ                                ]                                    (        1        )            
Accordingly, when dx/λ=α and dz/λ=β and g(θ, φ) represents the characteristic of each element antenna, the characteristic of the array antenna is represented by formula (2) below.
                              f          ⁡                      (                          θ              ,              ϕ                        )                          =                              ∑                          n              ,                              m                =                1                            ,              1                                      N              ,              M                                ⁢                                                    g                                  n                  ⁢                                                                          ⁢                  m                                            ⁡                              (                                  θ                  ,                  ϕ                                )                                      ⁢            exp            ⁢                          {                              -                                  j2π                  ⁡                                      [                                                                                            (                                                      n                            -                            1                                                    )                                                ⁢                        α                        ⁢                                                                                                  ⁢                        sin                        ⁢                                                                                                  ⁢                        θ                        ⁢                                                                                                  ⁢                        cos                        ⁢                                                                                                  ⁢                        ϕ                                            +                                                                        (                                                      m                            -                            1                                                    )                                                ⁢                        β                        ⁢                                                                                                  ⁢                        sin                        ⁢                                                                                                  ⁢                        ϕ                                                              ]                                                              }                                                          (        2        )            
Here, gnm(θ, φ) indicates the characteristic (e.g., gain) of an element antenna Anm, and λ indicates the wavelength of a carrier signal.
Sub-arrays can be obtained from the array antenna having the characteristic represented by formula 2.
FIG. 2 is a drawing used to describe an exemplary method of obtaining sub-arrays. Response matrices (A1 through A4) of sub-arrays A1 through A4 in FIG. 2 can be expressed by formula (3) below using phase offset matrices φ1 and φ2 and the response matrix A1 of the sub-array A1. In formula (3), a matrix A indicates matrices A1 through A4, collectively. However, the matrix A is merely a collection of submatrices and is different from an array response matrix.
                    A        ⁢                              Δ                          _              _                                ⁡                      [                                                                                A                    1                                                                                                                    A                    2                                                                                                                    A                    3                                                                                                                    A                    4                                                                        ]                          ⁢                              Δ                          _              _                                ⁡                      [                                                                                A                    1                                                                                                                                          A                      1                                        ⁢                                          Φ                      1                                                                                                                                                              A                      1                                        ⁢                                          Φ                      2                                                                                                                                                              A                      1                                        ⁢                                          Φ                      1                                        ⁢                                          Φ                      2                                                                                            ]                                              (        3        )            
Angle estimation may be performed by forming a least squares problem represented by formula (4) regarding a transformation matrix T and solving the least squares problem.
                              ɛ          ⁡                      (            T            )                          =                              min            T                    ⁢                                                                  E                -                AT                                                    2                                              (        4        )            
The phase offset matrices φ1 and φ2 (in formula (3)) are defined by diagonal matrices represented by formulas (5) and (6). In formulas (5) and (6), subscripts 1 through K indicate target identification numbers.Φ1=diag[exp(−j2πβ sin φ1), . . . ,exp(−j2πβ sin φK)]  (5)Φ2=diag[exp(−j2πα sin θ1 cos φ1), . . . ,exp(−j2πα sin θK cos φK)]  (6)
In formula (4), a matrix E is obtained by extracting submatrices (corresponding to the sub-arrays) from the results obtained by performing eigenvalue decomposition on a covariance matrix of an incoming signal matrix and by arranging the extracted submatrices in concord.
                    E        ⁢                              Δ                          _              _                                ⁡                      [                                                                                E                    1                                                                                                                    E                    2                                                                                                                    E                    3                                                                                                                    E                    4                                                                        ]                                              (        7        )            